Question

Identify the regression equation of X on Y and Y on X from the following equations:
$2x + 3y = 6$ and $5x + 7y - 12 = 0$

Answer 1

Hint: In order to solve this question we will know what is the regression equation and then for finding X on Y we will put the value of x from the first equation and we will find $by{x_1}$ and then we will find the value of y and further find$bx{y_1}$ and then we will multiply these both terms and compare it with ${r^2}$.

Complete step by step Solution:
For solving this question, first we will learn what is regression equation:
A regression equation is used in stats to find out what relationship, if any, exists between sets of data.
as it is given in question we need to find the regression equation of X on Y we will find the value of y in terms of x and put it in second equation:
$2x + 3y = 6$ so the value of y in terms of x will be:
$y = \dfrac{6}{3} - \dfrac{2}{3}x$
Now finding the $by{x_1} = - \dfrac{2}{3}$
Now from second equation we will find the value of x in terms of y:
$5x + 7y - 12 = 0$
Finding in terms of x
$x = \dfrac{{12 - 7y}}{5}$
Now we will be finding $bx{y_1} = - \dfrac{7}{5}$
If $2x + 3y = 6$ is X on Y and $5x + 7y - 12 = 0$ is Y on X
Then $x = \dfrac{{6 - 3y}}{2}$ and $y = \dfrac{{12 - 5x}}{7}$
Now we will find $bx{y_2} = - \dfrac{3}{2}$ and $by{x_1} = - \dfrac{5}{7}$
Now we will find the product of:
$bx{y_1} \times by{x_1} = \dfrac{{14}}{{15}} < 1$
As it less than 1 so it is correct;
Now we will check for:
$by{x_2} \times bx{y_2} = \dfrac{{15}}{4} > 1$

Since it is greater than one so this assumption is incorrect.

Note:
While solving these types of problems we should keep in mind that these questions are assumption based if the products of two bxy and byx which are equal to ${r^2}$ are less than one then the assumption is true while if the product is greater than 1 then the assumption is incorrect.